\section{Naive Stability Analysis of Numerical Integration Scheme}

This is just a simple verification regarding the stability of ordinary differential equations' (ODE system) numerical integration scheme. We try to give the simple proof for the simplest case, which is fixed time step integration for time invariant coefficients \textit{Dahlquist's test equations}. The equations or ODE system can be written in the following manner,

\begin{equation}\nonumber
\dot{\textbf{y}} = \textbf{Ay}
\end{equation}

And the statement we wanna prove is that if the eigenvalues of matrix \textbf{A} stay within the stability region of a numerical integration scheme after scaled by picked timestep, then that scheme is stable with that timestep when applied to solve this test ODE system.

Proof is straightforward and easy by making use of change of variables,

\begin{equation}\nonumber
\begin{split}
&\dot{\textbf{y}} = \textbf{Ay} = \textbf{V}^T\boldsymbol{\Lambda}\textbf{Vy}\\
\Rightarrow \textbf{V}&\dot{\textbf{y}} = \boldsymbol{\Lambda}\textbf{Vy} \Rightarrow \dot{\textbf{x}} = \boldsymbol{\Lambda}\textbf{x}
\end{split}
\end{equation}

The matrix $\boldsymbol{\Lambda}$ is the eigenvalue matrix of \textbf{A}. As we already decorelated the original ODE system into independent ODE equations, we can make use of our knowledge about the stability analysis for \textit{Dahlquist's test equation}, which then explains why all the timestep scaled eigenvalues need to stay inside the stability region. Here we did several simple but intuitive enough test cases by applying \textit{the 1st order Adams Bashforth} to the above problem. Here we provide several graph plotting the stability region versus the scaled eigenvalues.

\begin{figure}[ht]
\centering
\subfigure[]{
\includegraphics[width=0.4\textwidth]{img/ch12/1}}
\subfigure[]{
\includegraphics[width=0.4\textwidth]{img/ch12/4}}
\subfigure[]{
\includegraphics[width=0.4\textwidth]{img/ch12/3}}
\subfigure[]{
\includegraphics[width=0.4\textwidth]{img/ch12/2}}
\caption{Fig. a, b shows unstable timestep while Fig. c, d shows stable timestep}
\label{Fig.lable}
\end{figure}